[docs-infra] Implement search engine in v5 documentation page

• I have searched the issues of this repository and believe that this is not a duplicate.

Summary 💡

• In chrome, after using stack overflow, an search engine (managed with uri chrome://settings/searchEngines) is added to chrome omnibar, so when user types stack then with [tab] key a search on stack overflow can start directly. (when I search abcdef it will open the page https://stackoverflow.com/search?q=abcdef)
• I want to use this same function on https://mui.com/ maybe https://mui.com/search?q=abcdef can redirect to the homepage with the popup with abcdef typed in the pop-up.

Examples 🌈

• MDN: https://twitter.com/chantastic/status/1704924964804334064 Using DuckDuckGo Bang feature.
• Wikipedia:

Motivation 🔦

Usability